Yangians, Mirabolic Subalgebras, and Whittaker Vectors

We construct an element in a completion of the universal enveloping algebra of $\mathfrak{gl}_N$, which we call the Kirillov projector, that connects the topics of the title: on the one hand, it is defined using the evaluation homomorphism from the Yangian of $\mathfrak{gl}_N$, on the other hand, it gives a canonical projection onto the space of Whittaker vectors for any Whittaker module over the mirabolic subalgebra. Using the Kirillov projector, we deduce some categorical properties of Whittaker modules, for instance, we prove a mirabolic analog of Kostant's theorem. We also show that it quantizes a rational version of the Cremmer–Gervais $r$-matrix. As application, we construct a universal vertex-IRF transformation from the standard dynamical $R$-matrix to this constant one in categorical terms.